Following a solution to number 12, and this gives a summary stats that the sample main X bar is 35.1 and the sample standard deviation was a 8.7. Now notice it's a sample the esses sample standard deviation, and we don't know what sigma is, the population standard deviation. So since we don't know sigma, we have to use the tea interval instead of the z interval. Remember the less, you know, the more variable is going to be. So the t interval is what we're gonna use here, since we know s not sigma, and we're asked to find three things and then we talk about Um a little bit of conditions for inference on this last part, but let's look at a B&C because they kind of go together first, we're gonna find the 90% confidence interval winter and is 40. Then we're going to find the same level of confidence, 90% confidence interval, but this time the end is 100. So we increase that sample size And then this 3rd part We increase the confidence of them, but we keep the N equal to 40 again. And then we're going to talk a little bit about the conditions for inference. Now in the beginning it did not say anything about The population distribution. So normally it says assume that the population distribution is approximately normal. Well, here we don't need it to be normal because N. Is large enough. That magic number is 30. As long as your sample size is at least 30. It doesn't matter how your population is distributed. Um Because that's just the role as the central limit theorem world. Alright, so I'm gonna use technology for this because it cuts down. But you can use any excel. I'm gonna use the T. I. T. For but you can use any sort of technology Wish. And if you'd like to use the formula you should get the same answers. But I'm gonna use T. I. D. Four. So if you go to stat and then air over the tests it's this eighth option here the T. Interval. So you can scroll down or you can just click eight and make sure the summary stats is highlighted. So we don't have any data. Um we don't have a long list of data or anything. We're just giving the summary stats. So keep that highlighted and it asks for X. Bar. That's 35.1. S. Was 8.7 in was 40 at least for this first one and then the sea levels 90% of 900.9 For this first one. And we're gonna go and calculating this first band here. That's our interval. So 32.8 To 37.4 is r. t. interval. So 37 point I'm sorry 30-32 point 782 To 37.418. Okay so that's kind of like our base. So it's about uh we can kind of estimate maybe about five a part of 4 4.7 units apart. So just kind of keep that in the back of your mind 4.7 units apart. So let's see what happens whenever we increase that sample size to 100 and you may be able to guess by now. So if we go to the T. Interval again this time I'm just going to change this into 100 And let's see what happens. So we go from 36 I'm sorry 33.6-36.5. So let's write that down. Let's compare 33 .655. All the way up to 36 545 Okay. So notice that the lower bound gets higher and the upper bound gets lower. That means remember appear we were up to 4.7 here. It's You know about three actually a little bit less than 3-9. That interval um is Only 2.9 units so it's gotten narrower. So the next part of this question is how does the sample size or how does how do the degrees of freedom influence that margin of error? Or intern how does it influence the interval with? So as in increases or as the degrees of freedom increase, that margin of error is going to decrease. Which makes so making the confidence interval more narrow. Okay, so as you increase your in and you can kind of think about that standard air that? S over square devin, you're dividing by a bigger number which makes that margin of error smaller um and the smaller the margin of air then more narrow your confidence interval will be. So now let's look at the 98% confidence interval. But this time the ends back to 40. So if we go to stat Tests and it's that 8th 1 And this time we're going to bring that down to 40 and then we're gonna bring this one up to 98% of .98. We calculate and we get 31- 38. 31.8- 38.4. So it's right that down 31 point 763 two, points 437. So we're gonna compare this one to part a where it has the same sample size, just different confidence level. So we increase the confidence level and notice that we went from 4.7 units apart. And this is like 6.7 units apart. So it's gotten wider. All right. So What's that relationship? Well, as the confidence level increases, which is what we did here went from 90 to 98. That margin of air also increases because that intervals wider. So making the confidence interval wider. So the more confident you are, the wider and wider that interval must be. And then part D. It says what conditions of inference are needed if N equals 18. So if N equals 18 and I kind of sort already hinted on this, But since that end is less than 30, the population must be normally distributed since And is less than 30. Okay, so since N is less than 30, That population needs to be normal. Now. Normally they say that in the beginning but this time they didn't because these sample sizes are big enough, it doesn't matter. But if that in were equal to 18, 1 of those conditions for inference, if it is less than 30 is that the population comes from a normal distribution.